A semi-orthogonal path is a polygon inscribed into a given polygon such that the $i$-th side of the path is orthogonal to the $i$-th side of the given polygon. Especially in the case of triangles, the closed semi-orthogonal paths are triangles which turn out to be similar to the given triangle. The iteration of the construction of semi-orthogonal paths in triangles yields infinite sequences of nested and similar triangles. We show that these two different sequences converge towards the bicentric pair of the triangle's Brocard points. Furthermore, the relation to discrete logarithmic spirals allows us to give a very simple, elementary, and new constructions of the sequences' limits, the Brocard points. We also add some remarks on semi-orthogonal paths in non-Euclidean geometries and in $n$-gons.; Poluortogonalan put je poligonalna linija upisana u dani mnogokut takva da je $i$-ta stranica poligonalne linije okomita na $i$-tu stranicu danog mnogokuta. U slučaju trokuta, zatvoreni poluortogonalni putovi su trokuti slični danom trokutu. Iteracijom konstrukcije poluortogonalnih putova u trokutima dobivaju se beskonačni nizovi upisanih sličnih trokuta. Pokazujemo da ova dva različita niza konvergiraju prema bicentričnom paru Brocardovih točaka trokuta.Nadalje, veza s diskretnim logaritamskim spiralama omogućuje vrlo jednostavnu, elementarnu i novu konstrukciju limesa ovih nizova, Brocardovih točaka. Iznosimo i neke napomene o poluortogonalnim putovima kako u neeuklidskim geometrijama i tako i za $n$-kute.